EMET8002 Chap.3 When Regression Assumptions Fail
When Regression Assumptions Fail
Assumptions that protect the coefficients
Week 2 examines what happens when the assumptions behind regression fail.
Multicollinearity makes partial effects imprecise, and five routes to endogeneity, namely omitted variables, wrong functional form, measurement error, sample selection and simultaneous causality, bias the coefficients themselves.
Assumptions that protect the standard errors
Heteroskedasticity, correlated errors and non-normal errors leave coefficients unbiased but make standard errors inconsistent.
The chapter covers robust, HAC and clustered standard errors, outliers and leverage points, winsorising, and least absolute deviations with quantile regression.
Diagnosing one model in the Week 11 lab
The Week 11 lab asks you to test one multiple regression for normal residuals, constant variance, multicollinearity and linearity, then to add a per capita greenhouse gas variable with missing values.
Marks go to choosing the right test for each assumption, interpreting it in words, and explaining how the missing observations are handled rather than letting the software silently drop them.
What this chapter covers
- 01
Correlation cut-offs and the variance inflation factor
- 02
Conditional mean independence and bias
- 03
Omitted variables and the direction of bias
- 04
Functional form, measurement error and selection
- 05
Simultaneous causality
- 06
Heteroskedasticity and the Breusch-Pagan test
- 07
Clustered and HAC standard errors
- 08
Outliers, winsorising and least absolute deviations
Worked example · free
Signing the bias from omitted ability
- 1Ability sits in the error term and is correlated with education, so education is endogenous.
- 1The coefficient captures the direct effect of education plus an indirect effect of ability travelling through its correlation with education.
- 1Both links are positive, so the estimate is likely biased upwards.
Key terms
- Variance Inflation Factor
- A diagnostic for multicollinearity, with values above 10 taken as a warning.
- Heteroskedasticity
- A non-constant error variance that changes with a regressor or with the mean outcome.
- Clustered Standard Errors
- Standard errors that allow correlation within a cluster but not across clusters.
- Winsorising
- Replacing extreme values with a chosen percentile, such as the 5th and 95th.
- Least Absolute Deviations
- An estimator that minimises the sum of absolute residuals and targets the conditional median.
- Sample Selection Bias
- Bias created when the reason data are missing is related to the dependent variable.
When Regression Assumptions Fail FAQ
Does multicollinearity bias regression coefficients?
No. It inflates the variance of the affected coefficients, so their partial effects become imprecise and may look insignificant. The lecture's checks are pairwise correlations beyond 0.8, or 0.7 in some texts, and a variance inflation factor above 10.
What are the five causes of endogeneity in the lecture?
Omitted variables, misspecified functional form, measurement error in a regressor, missing data or sample selection related to the outcome, and simultaneous causality. Each makes a regressor correlated with the error term and biases its estimated coefficient.
When should I use robust or clustered standard errors?
Use heteroskedasticity-robust errors when the residual spread changes with the fitted values, and clustered errors when observations within a group, such as a school or a person over time, share shocks. For autocorrelated time series errors, HAC standard errors such as Newey-West are the tool.
How do I check for heteroskedasticity?
Plot the residuals against the fitted values and look for a spread that changes, such as a fan shape. The lecture names two formal tests, the Breusch-Pagan or Cook-Weisberg test and Cameron and Trivedi's decomposition. If the variance is not constant, respecify or transform the model, or use robust standard errors.
Should I drop outliers from my regression?
Not automatically. Correct genuine data-entry or calculation errors, then ask whether the remaining extreme values belong to the population you study. Theory decides whether to keep, trim or winsorise them, and least absolute deviations offers an estimator that is far less sensitive to them.
Assessment move
Sort every regression problem into one of two families before proposing a fix: problems that bias the coefficients, and problems that only distort the standard errors. Then attach the remedy the lecture lists for that case. Keep a one-page table for your research project listing each assumption, the check you ran and what you found; it becomes a ready-made robustness section.
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